where \(\langle \cdot ,\cdot \rangle \)  stands for the Hilbert-Schmidt inner product, i.e.,
\(\langle A,B \rangle ={\rm tr}(A^*B)\) .
The above definition of partial traces is implicit.
Assume that \(H=[H_{i,j}]_{i,j=1}^n\)  is an \(n\times n\)  block
matrix with \(H_{i,j}\in \mathbb {M}_k\) ,
the visualized version of the partial traces
is equivalently given in [1]} as
\( \mathrm {tr}_1 { H}=\sum \limits _{i=1}^n H_{i,i} ,\) 
